Uniform sets and complexity

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For a nonempty closed subset Ω of {0, 1}Σ, where Σ is a countably infinite set, let pΩ (S) {colon equals} # πS Ω be the complexity function depending on the nonempty finite sets S ⊂ Σ, where # denotes the number of elements in a set and πS : {0, 1}Σ → {0, 1}S is the projection. Define the maximal pattern complexity function pΩ* (k) {colon equals} supS ; # S = k pΩ (S) as a function of k = 1, 2, .... We call Ω a uniform set if pΩ (S) depends only on # S = k, and the complexity function pΩ (k) {colon equals} pΩ (S) as a function of k = 1, 2, ... is called the uniform complexity function of Ω. Of course, we have pΩ (k) = pΩ* (k) in this case. Such uniform sets appear, for example, as the partitions generated by congruent sets in a space with optimal positionings, or they appear as the restrictions of a symbolic system to optimal windows. Let Ω′ be the derived set (i.e. the set of accumulating points) of Ω and deg Ω {colon equals} inf {d ; Ω(d + 1) = 0{combining long solidus overlay}} with Ω(1) = Ω′, Ω(2) = (Ω′)′, .... We prove that for any nonempty closed subset Ω of {0, 1}N, where N = {0, 1, 2, ...}, such that deg (Ω {ring operator} ρ) < ∞ for some injection ρ : N → N, there exists an increasing injection φ{symbol} : N → N such that Ω {ring operator} φ{symbol} {ring operator} ψ = Ω {ring operator} φ{symbol} for any increasing injection ψ : N → N. Such a set Ω {ring operator} φ{symbol} is called a super-stationary set. Moreover, if deg (Ω {ring operator} ρ) = ∞ for any injection ρ : N → N, then pΩ* (k) = 2k (k = 1, 2, ...) holds. A uniform set Ω ⊂ {0, 1}Σ is said to have a primitive factor [Ω {ring operator} φ{symbol}] if there exists an injection φ{symbol} : N → Σ such that Ω {ring operator} φ{symbol} is a super-stationary set, where [Ω {ring operator} φ{symbol}] is the isomorphic class containing Ω {ring operator} φ{symbol}. Then, any uniform set has at least one primitive factor, and hence, any uniform complexity function is realized by the uniform complexity function of a super-stationary set. It follows that the uniform complexity function pΩ (k) is either 2k for any k or a polynomial function of k for large k. © 2008 Elsevier B.V. All rights reserved.




Kamae, T. (2009). Uniform sets and complexity. Discrete Mathematics, 309(12), 3738–3747. https://doi.org/10.1016/j.disc.2008.10.001

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